difference between connected domain and bounded domain what is the difference between connected domain and bounded domain? I am not able to understand their respective definitions which i found on the internet
 A: Since I do not exactly know your context, let's assume your domain is a subset of $\mathbb R^n$, equipped with the Euclidean norm.
Connectedness: There a different notions of connectedness, but a simple one is path-connectedness. Informally, a set is simply connected if any two points within the set can be connected by a continuous curves that entirely lies within the set. Very informally: A domain could be visualized by islands in the sea: A domain that is not connected would correspond to several islands - an ant cannot walk from one island to the other without getting wet, which illustrates the non-connectedness! A connected domain would correspond to a single island, no matter how its specific shape and size are.
Boundedness: On the other hand, a set is bounded if there exists some bound $M>0$ such that for every point $x$ in the set the norm is bounded by $M$, i.e. $\|x\| < M$. Informally: A domain is bounded if the distance between any two points cannot exceed a maximum. Given a sufficiently large sphere, you can fit the bounded domain completely inside the sphere. On the other hand, you can never find a sphere large enough to contain an unbounded domain. 


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*a water bottle defines a bounded domain in $\mathbb R^3$: it will entirely fit into a sphere with radius $>$ 1 meter (to be one the safe side). The distance between any two points cannot exceed a certain distance (which is probably slightly more than the height of the bootle).

*In real life, unbounded domains are hard to find. If we could guess that the universe was infinite, it would be an example for an unbounded domain: No matter how large you set a threshold, you could always find a star that is even further away from our sun. In mathematics, unbounded domains in $\mathbb R^n$ contain a sequence whose norm diverges to $\infty$. A simple example is the interval $[0,\infty)$, which is unbounded: For every threshold $M$ it contains numbers whose distance from $0$ is larger than $M$.


Connectedness and boundedness are completely unrelated properties. However, often (e.g. in the context of PDEs, ...) both of them are imposed on domains.
